On Codes over Eisenstein Integers

Abstract

We propose constructions of codes over quotient rings of Eisenstein integers equipped with the Euclidean, square Euclidean, and hexagonal distances as a generalization of codes over Eisenstein integer fields. By set partitioning, we effectively divide the ring of Eisenstein integers into equal-sized subsets for distinct encoding. Unlike in Eisenstein integer fields of prime size, where partitioning is not feasible due to structural limitations, we partition the quotient rings into additive subgroups in such a way that the minimum square Euclidean and hexagonal distances of each subgroup are strictly larger than in the original set. This technique facilitates multilevel coding and enhances signal constellation efficiency.